Using small sharp knife, cut the stem from top of each fig, then cut straight down from the top to the center to make ¾-inch-deep, ½-inch-long slit in each fig. Fill the slit in each fig with a cube of feta cheese, then press the opening closed. Place 1 slice of prosciutto on a work surface; top with a sage leaf. Place a stuffed fig at 1 end and roll up to enclose the fig. Secure with a toothpick. Repeat with the remaining prosciutto, sage, and figs. Arrange on a plate, cover tightly with plastic wrap, and chill until ready to cook.
COOK THE SOUVLAKI:
Remove the plate of appetizers from the refrigerator. Let them stand at room temperature 1 hour before continuing. Pour enough oil into a large skillet to cover the bottom. Heat over medium-high heat. Add the figs. Cook until the prosciutto begins to crisp, about 2 minutes per side. Transfer to a large platter. Serve warm.
Can be made 1 day ahead (through step 1). Continue with step 2 when ready to serve.